A *spanning subgraph of (a necessarily connected) \(G\)  is a connected subgraph that contains all the vertices of \(G\) .
We call a set \({\hspace{0.2pt}\texttt {E}}^{\prime }\subseteq {\hspace{0.2pt}\texttt {E}}\)  of edges spanning if \(G[{\hspace{0.2pt}\texttt {E}}^{\prime }]\)  is spanning, equivalently if it is connected.
As customary in circuit literature (see e.g. [1]}, [2]}, [3]}), a *tree \(\texttt {T}\)  of \(G\)  always means a spanning tree, namely an acyclic spanning subgraph of \(G\) .
Its cotree \({\texttt {T}^*}\)  is the subgraph of \(G\)  obtained by removing the edges of \(G\)  that are in \(\texttt {T}\) .
We usually identify a tree with its set of edges, and the cotree similarly.
In that sense \({\texttt {T}^*}\)  is the complement of \(\texttt {T}\) , i.e.,
\({\texttt {T}^*}= {\hspace{0.2pt}\texttt {E}}\setminus \texttt {T}\) .
As in [2]} we call tree edges twigs and cotree edges *links.
